Proof of Theorem  REF
The coherence measure \(C_f\)  in Eq. (REF ) corresponds one-to-one to the real symmetric concave function \(f\in {\mathfrak {{F}}}\setminus \lbrace 0\rbrace \)  [1]}, [2]}. The entanglement measure \(E_f\)  in Eq. (REF ) also corresponds one-to-one to the real symmetric concave function \(f\in {\mathfrak {{F}}}\setminus \lbrace 0\rbrace \)  [3]}. Therefore, the coherence measure \(C_f\)  in Eq. (REF ) and the entanglement measure \(E_f\)  in Eq. (REF ) are in one-to-one correspondence. As the least concave majorant extension of the coherence measure \(C_f\)  and entanglement measure \(E_f\) , the coherence of assistance \({C_a}\)  and the entanglement of assistance \(E_a\)  are also in one-to-one correspondence.
